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METHOD:PUBLISH
BEGIN:VEVENT
DTSTAMP:20221031T095545Z
DTSTART:20221115T150000Z
DTEND:20221115T160000Z
SUMMARY:Manchester Number Theory Seminar - Francesco Ballini
UID:{http://www.columbasystems.com/customers/uom/gpp/eventid/}uff-l8vgb05
8-bvb378
DESCRIPTION:Speaker: Francesco Ballini (University of Oxford)\n\nTitle: H
yperelliptic continued fractions\n\nAbstract: We can define a continued
fraction for formal series f(t)=\\sum_{i=-\\infty}^d c_it^i by repeatedl
y removing the polynomial part\, \\sum_{i=0}^d c_it^i\, (the equivalent
of the integral part) and inverting the remaining part\, as in the real
case. This way\, the partial quotients are polynomials. Both the usual r
eal continued fractions and the polynomial continued fractions carry pro
perties of best approximation. However\, while for square roots of ratio
nals the real continued fraction is eventually periodic\, such periodici
ty does not always occur for \\sqrt{D(t)}. The correct analogy was found
by Abel in 1826: the continued fraction of \\sqrt{D(t)} is eventually p
eriodic if and only if there exist nontrivial polynomials x(t)\,y(t) suc
h that x(t)^2-D(t)y(t)^2=1 (the polynomial Pell equation). Notice that t
he same holds also for square root of integers in the real case. In 2014
Zannier found that some periodicity survives for all the \\sqrt{D(t)}:
the degrees of their partial quotients are eventually periodic. All thes
e statements are strongly related to geometry and they are based on the
study of the Jacobian of the curve u^2=D(t). We give a brief survey of t
he theory of polynomial continued fractions and their interplay with Jac
obians.\n\nRoom: Frank Adams 1
STATUS:TENTATIVE
TRANSP:TRANSPARENT
CLASS:PUBLIC
LOCATION:Frank Adams 1\, Alan Turing Building\, Manchester
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